A huge idea in advanced mathematics is that you can add objects that aren’t numbers. People often don’t learn about this surprising concept until college, if ever, and the sudden abstraction can come as a shock. Yet even little kids enjoy playing with abstract objects, if given the right materials.

The card game Dot, Dot, Poof! teaches kids about a system where you can add cards of colorful dots. It’s a fun game of visual patterns with game play similar to SET, but accessible to even younger kids. Along the way, it also invites exploration into the basic concepts of linear algebra.

## The Addition Rule

The cards show red, yellow, green, blue, and purple dots. The colors always appear in the same order.

The cards follow a new rule for “adding”:

**To add two cards, find the color dots that appear on one, ****but not both, of the two cards. **

This breaks down as:

- If the color appears on both cards, it disappears in the result.

(dot + dot = poof!) - If the color appears on exactly one card, it is included in the result.

(dot + blank = blank + dot = dot) - If the color appears on neither card, it is still not included in the result

(blank + blank = blank)

For example, the orange and purple dots are on both cards here, and so those colors disappear in the result:

You can try this example. Click on the image to check your answer.

## Triad!

There are several games to play with the Dot, Dot, Poof! cards. (Download them now if haven’t yet: color / grayscale) The first game practices the addition rule for the cards.

Shuffle the cards. Lay 12 cards in a 3 x 4 grid with the small gray arrow at the top right of each card.

Players race to find triads of cards where one card is the sum of the other two. The player who spots such a triplet calls “Triad!” and picks up the three cards. The other players check that the triad is valid.

Lay out three more cards to replace each triad taken. If at any point the players agree there are no triads among the cards laid out, lay out three more cards.

The game continues until there are no valid triads remaining. The player with the most triads is the winner.

## Combo! Part 1

The next game develops the idea that cards in the system are sums of other cards. In the language of linear algebra, each card is a linear combination of other cards. (In other systems you would need to allow for multiples of cards, but the “dot + dot = poof” rule creates a simpler system here.)

Shuffle the cards. Lay 12 cards in a 3 x 4 grid with the small gray arrow at the top right of each card.

Players race to find one card that is the sum of other cards displayed. Any number of cards may be used. When a player spots such a card, they call “Combo!” and pick up that one card. They point out the other cards for the other players to check, but leave them on the table.

Lay out one new card to replace the card taken. If at any point the players agree that no card is the sum of any other cards, lay out three more cards.

Continue until there are no more cards to take. (Tip: There should be five cards remaining.) The winner is the one who took the most cards.

## Combo! Part 2

This game immediately follows on Part 1, and it gets to the core of linear algebra.

The five cards remaining in Part 1 are special. Notice that none of the five cards is sum of any of the others, which means that the cards are independent. (Again, multiples of cards are not needed because of the “dot + dot = poof” rule.)

Now in Part 2 of the game, you will show that you can use the five cards to get any other card in the system. In linear algebra language, this means that the five cards remaining in Part 1 span the system of cards. Because the five cards are independent and span the whole system, they form a basis for the system. These five cards are special because they can make any other card in the system, while no smaller set of cards would work.

Leave the five cards from Part 1 on the table. Each player keeps their pile of cards face down in front of them.

Players take turns flipping over a card from their pile. Players race to find which cards among the five cards add up to the flipped target card. Any number of the five cards may be used.

The player who finds such a combination calls “Combo!” They point to their proposed combination for the other players to check. If they are right, they take the one target card, leaving the five cards on the table.

Play until all the draw piles are empty. The winner is the one who has the most cards at the end.

Note: To make Part 2 more challenging, add restrictions to the rules in Part 1 so that you don’t always get the simplest basis of cards left at the end. For example, you can include the additional challenge that players may not take a card if it has more dots than any of the cards used in the sum.

## Asking Good Questions

Lots of questions should come up naturally while playing. Here are some questions you can propose or listen for:

Does order matter when adding two cards?

In the integer number system, 0 plus any number is the same number. What is the equivalent of 0 in the card system?

In the integer number system, every number has an opposite so that the number and its opposite add to 0. What is the opposite of a card in the card system?

What is left at the of a game of Triad? Is it always the same cards? Is it always the same number of cards?

How many cards are left at the end of Combo! Part 1? Why?

Is every card a sum of cards on the table in Combo! Part 2? Why?

What will you and your little (or big!) ones discover? Please share your discoveries in the comments!

## Variations and Extensions

Finally, here are some further directions for exploring your system of cards. What other ideas do you have? Please share them in the comments!

- Remove a random card from the deck, without looking. Can you figure out what card is missing?

- When playing Combo! Part 1, include an additional challenge: Players may not take a card if it has more dots than any of the cards used in the sum.

- Record the cards left at the end of Combo! Part 1. How many different sets of cards can you find?

- Play a shorter game by setting aside the cards with purple dots, leaving a deck of 15 cards. Then how many cards are left at the end of Combo! Part 1?
- Investigate a smaller system of one, two, or three colors by setting aside the extra cards. How many cards are left at the end of playing the games with these systems? Can you make an addition table for each system?

## Footnote: a Source and Inspiration

I developed Dot, Dot, Poof! from ideas that mathematician and educator Zoltan Paul Dienes wrote about in his book I Will Tell You Algebra Stories You Have Never Heard Before. It’s an astounding book, and entirely true to its name. Dienes believed young children could learn and enjoy abstract math, such as linear algebra and group theory, through games and even dances. While Professor Dienes is best known for popularizing the use of base ten blocks (which are called Dienes blocks in the UK), his work on teaching abstract math to very young children deserves to be known better.